2 December 2023 Geometry in mathematics focuses on the study of geometric figures, either in three-dimensional space or on a two-dimensional plane. The cube is a geometric figure that can be described as a parallelepiped with bases formed by parallelograms.

Like any other parallelepiped, the cube has four diagonals that meet at its midpoint. Besides, has eight vertices, sixteen edges and six pairs of faces that are equal and parallel. It is important to note that the cube is the only polyhedron that has all its faces in the form of squares. Next, we explain in more depth what the cube is, its meaning and its characteristics.

## Concept of a cube in geometry

As a concept in geometry, a cube is a geometric body that is made up of six square-shaped faces. Each of these faces is congruent to each other, which means that they have the same measure and shape. The faces of the cube They are arranged in parallel pairs. and each face has four sides.

Due to their unique characteristics, cubes can be classified into different groups. These include the Platonic solids, convex polyhedra, parallelograms, hexagons, and prisms. Each of these groups is defined by the particular geometric properties of the cube. It is important to note that The cube is a three-dimensional object. and therefore has other interesting properties, such as its diagonals, vertices, edges, and volume.

A cube is a geometric solid that has six congruent square faces, and each interior angle is a right angle. The size of all the faces is equal, which means that all the edges of the cube have the same length. Besides, a regular hexagon is a polygon It has six sides and six equal angles.

The cube has interesting geometric properties, such as its four orthogonal projections centered in a vertex, edge or face, that fit perfectly to your figure. Furthermore, the cube can be represented as a spherical polyhedron and projected onto the plane using the stereographic projection, which preserves angles, but not areas or lengths. In this projection, the lines on the sphere resemble arcs of circles in the plane. Regarding its topology, the cube has properties such as its connectedness, border, open, exterior, convexity, and interior.

## What are the features and parts of a cube?

Below is a list of features and parts of a cube:

• Faces: The cube has six congruent square faces.
• Edges: The cube has 12 equal edges of equal length.
• Vertices: The cube has 8 vertices where three edges meet.
• Angles: Each face of the cube is at right angles to two adjacent faces. The angles at each vertex measure 90 degrees.
• Symmetry: The cube is symmetric in all directions.
• Cuts: Any straight cut through the center of the cube results in two equal sections.
• Surfaces: The faces are flat surfaces that form the sides of the cube.

### What is the edge of a cube?

Edges are the line segments that connect the vertices of the figure. Since the cube has six faces, it has a total of 12 edges. Each edge connects two vertices and is formed by the intersection of two faces. All the edges of the cube have the same length, since all faces are congruent squares. Therefore, the length of an edge can be calculated by dividing the length of a diagonal of a face by the square root of 2. The edges of the cube are important in calculating its volume, since the volume of a cube can be find by cube the length of an edge.

## How many faces does a cube have?

A cube has six faces. All faces are congruent squares, that is, they have the same shape and size. Each face of the cube is connected to four edges and two other squares. Opposite faces of the cube are parallel and have the same shape and size. In general, the cube is a regular polyhedron with six faces, where each face is a congruent square.

## What are the formulas for a cube?

The formulas of a cube they are usually:

• Volume: The volume of a cube can be calculated using the formula V = a^3, where a is the length of either side of the cube.
• Surface area: The surface area of ​​a cube can be calculated using the formula A = 6a^2, where a is the length of either side of the cube.
• Diagonal Length: The length of the diagonal of a cube can be calculated using the formula d = a√3, where a is the length of any side of the cube.
• Edge length: The edge length of a cube can be calculated using the formula a = V^(1/3), where V is the volume of the cube.
• One-sided diagonal: The diagonal of a face of a cube is equal to the diagonal of a square, which can be calculated with the formula d = l√2, where d is the diagonal and l is the length of one side of the square. In a cube, all the faces are squares, so all the diagonals of the faces are the same length.
• Cube diagonal: The diagonal of the cube is the straight line that goes from one vertex to another opposite vertex passing through the center of the cube. It can be calculated with the formula D = L√3, where D is the diagonal of the cube and L is the length of one side of the cube.

### What is the volume of a cube?

The volume of a cube is the amount of space it occupies in three-dimensional space. Mathematically, it is calculated by multiplying the value of the length of one of the edges by itself three times, that is, raising the length to the cube. The formula for the volume of a cube uses the following data:

Where ‘V’ represents the volume of the cube and ‘a’ is the length of one of its edges. The result is expressed in cubic units, such as cm^3, m^3, etc. For example, if the side of the cube measures 5 centimeters, the calculation would be:

Volume = 5cm x 5cm x 5cm = 125cm³

Therefore, the volume of the cube is 125 cubic centimeters. It is important to note that the unit of measure is raised to the cube, since it is a volume.

### What is the lateral area of ​​a cube?

The lateral area of ​​a cube is the sum of the areas of the six faces of the cube, excluding the areas of the bases. That is, it is the sum of the areas of the four lateral faces of the cube. Each lateral face of the cube is a square with a side equal to the length of one edge of the cube. Therefore, the lateral area of ​​the cube can be calculate by multiplying the length of an edge by the height of a lateral face of the cube, which is also equal to the length of an edge. Mathematically, the lateral area of ​​a cube can be expressed as:

Side area of ​​cube = 4 x (length of edge)^2. For example, if we have a cube with an edge length of 5 cm, We can calculate its lateral area as follows:

• Lateral area of ​​the cube = 4 x (5 cm)^2
• Lateral area of ​​the cube = 4 x 25 cm^2
• Lateral area of ​​the cube = 100 cm^2

Therefore, the lateral area of ​​this cube would be 100 square centimeters.

The study of the cube in geometry is important for several reasons:

• Fundamentals of geometry: The cube is one of the simplest and most fundamental objects in three-dimensional geometry. Studying its basic properties and characteristics lays the foundations for the study of other geometric solids more complex.
• Concepts of volume and area: The cube is one of the few geometric solids that has a simple and precise formula to calculate both its volume and its area. The study of the cube allows students to better understand these concepts and develop skills in calculation and measurement.
• Development of spatial thinking: The cube is a three dimensional object which requires the ability to visualize and manipulate in space. Study of the cube can help students develop skills in spatial thinking and visual perception.
• Applications in everyday life: The cube is used in numerous applications in everyday life, from the construction of buildings to the manufacture of objects and machinery. The study of the cube and of geometry in general can help students better understand the world around them and develop useful skills in fields such as engineering and architecture.